# Tagged Questions

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### Calculation Of Integral Related To Sequence

Let's evaluate the following integral. Many trials but no success. $$\int_{-\pi}^{\pi}\dfrac{\sin nx}{(1+\pi^{x})\sin x}dx$$
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### Integration using summation

How do you integrate $\sqrt{x}$ from an arbitrary constant $a$ to another $b$ by summation ?
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### Evaluating a slow sum

In my integration adventures, I came across this sum which I could not simplify: $$\sum_{n=1}^{\infty}\frac{(-1)^{n}\log(2n+1)}{2n+1}$$ Wolfram seems to believe the sum diverges and is not of much ...
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### How to evaluate $\displaystyle\int_0^1\frac{\log^2(1+x)}x\mathrm dx$?

The definite integral $$\int_0^1\frac{\log^2(1+x)}x\mathrm dx=\frac{\zeta(3)}4$$ arose in my answer to this question. I couldn't find it treated anywhere online. I eventually found two ways to ...
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### Calculate : $\int_1^{\infty} \frac{1}{x} -\sin^{-1} \frac{1}{x}\ \mathrm{d}x$

Find : $\displaystyle \int_1^{\infty} \frac{1}{x} -\sin^{-1} \frac{1}{x}\ \mathrm{d}x$. I've done some work but I've got stuck, you may try to help me continue or give me another way , in both cases ...
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### A rational function integral

How to evaluate : $$\int_{0}^{1}{\frac{{{x}^{a-1}}}{1+{{x}^{b}}}}\text{d}x,\ \ \ a,\ b\in {{\mathbb{N}}^{+}}$$
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### A integral with hyperbolic function

Evaluate : $$\int_{0}^{2{\rm arccosh\,} \pi }{\frac{\text{d}x}{1+\frac{{{\sinh }^{2}}x}{{{\pi }^{4}}}}}$$
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### How to calculate a infinite Riemann sum

I am working on this assignment and I got a little stuck up with this. I got some ideas but I not at all sure if they are correct. So I am hoping to get opinion from someone in here please. How to ...
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### Prove that $\int_{0}^{1} \ln\left(\frac{1-a x}{1-a}\right) \frac{1}{\ln x} \mathrm{dx} = -\sum_{k=1}^{\infty} a^{k} \frac{\ln(1+k)}{k}, \space a<1$

Prove that $$\int_{0}^{1} \ln\left(\frac{1-a x}{1-a}\right) \frac{1}{\ln x} \mathrm{dx} = -\sum_{k=1}^{\infty} a^{k} \frac{\ln(1+k)}{k}, \space a<1$$ I find this question rather troublesome since ...
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### Compute $\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$

Compute the limit $$\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$$
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### Show that the integral of this rational function is equal to an infinite alternating harmonic series

One of my friends gave me the following question from his review, I have little experience to dealing with these types of questions in Analysis so if you could help us just to get started it would be ...
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### expansion of $\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt$ in inverse powers of $p$

This question relates to this answer I gave to a question about the integral $$\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt\;.$$ I derived an expansion in inverse powers of $p$ and then ...
I'm trying to solve the following integral: $$\int_{-1}^{1}C_{n_1-l_1}^{l_1+1}(x)C_{n_2-l_2}^{l_2+1}(x)C_{n_3-l_3}^{l_3+1}(x)(1-x^2)^{(l_1+l_2+l_3+1)/2}dx$$ Where $C_{n}^{\lambda}(x)$ is a ...
This is the last part of a three part problem on characteristic functions, and it's been driving me crazy over the last few days. Any help would be most appreciated. $X_1,X_2, \ldots, X_n$ are ...